Re: [Algorithms] solving for multiple matrices
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Re: [Algorithms] solving for multiple matrices
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From: Bill B. <wb...@gm...> - 2009-09-18 16:44:44
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That was my first thought too, but you don't need to a *Cholesky* decomp on C and D. Another kind of factorization will do, like a diagonalization -- http://en.wikipedia.org/wiki/Square_root_of_a_matrix --bb On Fri, Sep 18, 2009 at 7:51 AM, Andras Balogh <and...@gm...> wrote: > Hmm, I just looked up Cholesky decomposition, and it says you can only do > that if the matrix is symmetric and positive definite. Since the original > definition of C = A1^-1 * A2, where A1 and A2 both contain arbitrary > translation and rotation, I don't think C will be symmetric. Or am I > missing something? > > Andras > > > On Fri, 18 Sep 2009 01:58:41 -0600, Gino van den Bergen > <gin...@gm...> wrote: > >> If Y can be considered a rotation then Y is orthogonal and thus Y^-1 = >> Y^T, in which case this equation can be solved through Cholesky >> decomposition: >> >> For C = Y^T * D * Y, let's decompose >> >> C = U^T * U, and >> D = V^T * V >> >> then >> >> U^T U = Y^T * V^T * V * Y >> = (V * Y) ^T * (V * Y) >> >> this gives U = V * Y >> >> and thus Y = V^-1 * U >> >> Basically you are taking the square root of a matrix. >> >> Hope this helps, >> >> Gino >> >> >> >> Andras Balogh wrote: >>> Both X and Y matrices represent a simple translation and rotation. For >>> the >>> case of the Y matrix, the translation part will likely to be very small, >>> so I could probably pretend it's only rotation. >>> >>> What I would really like though, is to find a solution, where I could >>> use >>> more than two equations, eg: >>> A1 = X * B1 * Y >>> A2 = X * B2 * Y >>> A3 = X * B3 * Y >>> ... >>> An = X * Bn * Y >>> >>> And then compute a least squares solution from this over-constrained >>> system. >>> BTW, when I said that I'm looking for an analytical solution, I just >>> meant >>> something that is not based on an iterative approach. As long as I can >>> get >>> to a part where I have to solve a large system of over-constrained >>> linear >>> equations, I'm home. Unfortunately, I don't know how to make this >>> linear.. >>> >>> >>> Andras >>> >>> >>> >>> >>> On Thu, 17 Sep 2009 16:32:25 -0600, Jon Watte <jw...@gm...> wrote: >>> >>> >>>> Andras Balogh wrote: >>>> >>>>> Then it becomes: >>>>> C = Y^-1 * D * Y >>>>> >>>>> Now, how do I solve this for Y? This form lookes strangely familiar, >>>>> but I >>>>> cannot figure out what to do from here (wish I knew how to Google this >>>>> ;). >>>>> Hopefully there's an analytic solution to this. Any ideas? >>>>> >>>>> >>>>> >>>> That's the formula for applying a rotation in the reference frame of >>>> another rotation. >>>> >>>> Do you know anything more about these matrices than that they are >>>> matrices? Are they supposed to contain no scale? No translation? If you >>>> can formulate them as quaternions, writing out the analytical answer is >>>> a lot simpler :-) >>>> >>>> Sincerely, >>>> >>>> jw >>>> >>>> >>>> >>>> >>> >>> >>> >>> ------------------------------------------------------------------------------ >>> Come build with us! The BlackBerry® Developer Conference in SF, CA >>> is the only developer event you need to attend this year. Jumpstart your >>> developing skills, take BlackBerry mobile applications to market and >>> stay >>> ahead of the curve. Join us from November 9-12, 2009. Register >>> now! >>> http://p.sf.net/sfu/devconf >>> _______________________________________________ >>> GDAlgorithms-list mailing list >>> GDA...@li... >>> https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list >>> Archives: >>> http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list >>> >> >> >> ------------------------------------------------------------------------------ >> Come build with us! The BlackBerry® Developer Conference in SF, CA >> is the only developer event you need to attend this year. Jumpstart your >> developing skills, take BlackBerry mobile applications to market and stay >> ahead of the curve. Join us from November 9-12, 2009. Register >> now! >> http://p.sf.net/sfu/devconf >> _______________________________________________ >> GDAlgorithms-list mailing list >> GDA...@li... >> https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list >> Archives: >> http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list > > > > ------------------------------------------------------------------------------ > Come build with us! The BlackBerry® Developer Conference in SF, CA > is the only developer event you need to attend this year. Jumpstart your > developing skills, take BlackBerry mobile applications to market and stay > ahead of the curve. Join us from November 9-12, 2009. Register now! > http://p.sf.net/sfu/devconf > _______________________________________________ > GDAlgorithms-list mailing list > GDA...@li... > https://lists.sourceforge.net/lists/listinfo/gdalgorithms-list > Archives: > http://sourceforge.net/mailarchive/forum.php?forum_name=gdalgorithms-list > |
